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arXiv:cond-mat/0201221

Transfer matrix and Monte Carlo tests of critical exponents in lattice models

J. Kaupužs

Published Jan 14, 2002• 0 citations
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Abstract

The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.

DOI
10.48550/arxiv.cond-mat/0201221

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The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
We have tested the leading correction-to-scaling exponent omega in O(n)-symmetric models on a three-dimensional lattice by analysing the recent Monte Carlo (MC) data. We have found that the effective critical exponent, estimated at finite sizes of the system L and L/2, decreases remarkably within the range of the simulated L values. This shows the incorrectness of some claims that omega has a very accurate value 0.845(10) at n=1. A selfconsistent infinite volume extrapolation yields row estimates omega=0.547, omega=0.573, and omega=0.625 at n=1, 2, and 3, respectively, in approximate agreement with the corresponding exact values 1/2, 5/9, and 3/5 predicted by our recently developed GFD (grouping of Feynman diagrams) theory. We have fitted the MC data for the susceptibility of 3D Ising model at criticality showing that the effective critical exponent eta tends to increase well above the usually accepted values around 0.036. We have fitted the data within [L;8L], including several terms in the asymptotic expansion with fixed exponents, to obtain the effective amplitudes depending on L. This method clearly demonstrates that the critical exponents of GFD theory are correct (the amplitudes converge to certain asymptotic values at L tending to infinity), whereas those of the perturbative renormalization group (RG) theory are incorrect (the amplitudes diverge). A modification of the standard Ising model by introducing suitable "improved" action (Hamiltonian) does not solve the problem in favour of the perturbative RG theory.
Transfer matrix calculations of the critical two-point correlation function in 2D Ising model on a finite-size [Formula: see text] lattice with periodic boundaries along 〈11〉 direction are extended to L = 21. A refined analysis of the correlation function in 〈10〉 crystallographic direction at the distance r = L indicates the existence of a nontrivial finite-size correction of a very small amplitude with correction-to-scaling exponent ω < 2 in agreement with our foregoing study for L ≤ 20. Here we provide an additional evidence and show that amplitude a of the multiplicative correction term 1 + aL-ωis about -3.5·10-8if ω = 1/4 (the expected value). We calculate also the susceptibility for L ≤ 18 in order to compare our numerical estimates for the constant background contribution with the known very precise value and to look for possible nontrivial corrections to scaling. The numerical analysis reveals a perfect agreement for the background term, as well as shows that the nontrivial correction term, detected by our analysis in the correlation function, likely cancels in the susceptibility.
Thermal and chiral critical exponents of the fully frustrated XY model on a square lattice are obtained from a finite-size scaling analysis of the free energy of chiral domain walls. Data were obtained by extensive Monte Carlo transfer-matrix computations for infinite strips of widths up to 14 lattice spacings. Two transfer matrices were implemented, one for each of two principal lattice directions. The results of both are consistent, but the critical exponents differ significantly from the pure Ising values. This is in agreement with other recent Monte Carlo simulations. Our results also support the identification of the critical behavior of this model with that along the line of transitions of simultaneous ordering or becoming critical of Ising and planar rotor degrees of freedom in the XY-Ising model studied recently.
We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-size-scaling analysis. Using a minimal set of assumptions we find that the spectrum of correction-to-scaling exponents is consistent with that of the exactly solved Ising model, and that the critical exponents and correlation-length amplitudes closely follow the relation predicted by conformal invariance. Assuming that these spectra are exactly identical, and conformal invariance, we determine the critical point, the conformal anomaly, and the temperature and magnetic exponents with numerical margins of 10(-11) or less. These results are in a perfect agreement with the exactly known Ising universal parameters in two dimensions. In order to obtain this degree of precision, we included system sizes as large as feasible, and used extended-precision floating-point arithmetic. The latter resource provided a substantial improvement of the analysis, despite the fact that it restricted the transfer-matrix calculations to finite sizes of at most 34 lattice units.